Simplify the following expression and state the condition under which the simplification is valid. You can assume that $k \neq 0$. $x = \dfrac{4(5k - 7)}{-6} \div \dfrac{3k(5k - 7)}{7k} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{4(5k - 7)}{-6} \times \dfrac{7k}{3k(5k - 7)} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 4(5k - 7) \times 7k } { -6 \times 3k(5k - 7) } $ $ x = \dfrac{28k(5k - 7)}{-18k(5k - 7)} $ We can cancel the $5k - 7$ so long as $5k - 7 \neq 0$ Therefore $k \neq \dfrac{7}{5}$ $x = \dfrac{28k \cancel{(5k - 7})}{-18k \cancel{(5k - 7)}} = -\dfrac{28k}{18k} = -\dfrac{14}{9} $